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INVARIANTIVE  CHARACTERIZATIONS  OF 

LINEAR  ALGEBRAS  WITH  THE 

ASSOCIATIVE  LAW  NOT 

ASSUMED 


A  DISSERTATION 

Submitted  to  the  Faculty  op  the  Ogden  Graduate  School 
OF  Science  I^f  Candidacy  for  the  Degree 
of  Doctor  of  Philosophy 

department  of  mathematics 


j 


BY 

CYRUS  COLTON  MACDUFFEE 


Private  Edition 

Distributed  By 

THE  UNIVERSITY  OF  CHICAGO  LIBRARIES 

CHICAGO,  ILLINOIS 


Reprinted  from  The  Transactions  of  the  American  Mathematical  Society, 
Vol.  23,  No.  2.  March  1923 


Zbc  Tantversiti?  of  Cbicago 


INVARIANTIVE  CHARACTERIZATIONS  OF 

LINEAR  ALGEBRAS  WITH  THE 

ASSOCIATIVE  LAW  NOT 

ASSUMED 

A  DISSERTATION 

Submitted  to  the  Faculty  of  the  Ogden  Graduate  School 
OF  Science  in  Candidacy  for  the  Degree 
OF  Doctor  of  Philosophy 

department  op  mathematics 


BY 

CYRUS  COLTON  MACDUFFEE 


Private  Edition 

Distributed  By 

THE  UNIVERSITY  OF  CHICAGO  LIBRARIES 

CHICAGO,  ILLINOIS 


Reprinted  from  The  Transactions  of  the  American  Mathematical  Society, 
Vol.  23,  No.  2.  March  1923 


EXCK.---.fHI 


>  »\  »  » 


INVARIANTIVE  CHARACTERIZATIONS  OF  LINEAR  ALGEBRAS 
WITH  THE  ASSOCIATIVE  LAW  NOT  ASSUMED* 

BY 

CYRUS  COLTON  MACDUFFEE 

I.  Introduction 

We  consider  linear  algebras  in  \vhich  neither  the  commutative  nor  the  asso- 
ciative law  of  multiplication  is  assumed,  and  whose  coordinates  and  constants 
of  multiplication  are  numbers  of  a  general  field  F.  A  rational  integral  invariant, 
or  covariant,  is  a  rational  integral  function  of  the  constants  of  multiplication, 
or  of  the  constants  of  multiplication  and  the  coordinates  of  the  general  number, 
which  has  the  invariantive  property  under  the  total  group  of  linear  homogeneous 
transformations  on  the  units.  If  an  invariantive  function  also  actually  involves 
the  units,  it  has  been  called  a  vector  covariant  by  Professor  O.  C.  Hazlett.f 
who  shows  that  every  rational  integral  vector  covariant  can  be  obtained  as  a 
covariant  of  the  general  number  of  the  algebra  and  a  fundamental  set  of  ordinary 
covariants. 

In  Section  II  of  this  article,  it  is  shown  how  vector  covariants  may  be  formed 
directly  from  the  constants  of  multiplication  without  assuming  the  knowledge 
of  any  ordinary  covariants  or  invariants.  To  do  this,  the  notion  is  introduced 
of  a  hypercomplex  determinant  whose  elements  obey  neither  the  associative  nor 
the  commutative  law  of  multiplication,  and  a  few  simple  properties  of  such  hyper- 
complex determinants  are  derived.  From  the  vector  covariants  and  the  charac- 
teristic determinants  of  the  algebra,  ordinary  relative  invariants  may  easily  be 
found. 

In  Section  III  the  linear  algebra  in  three  units,  one  of  which  is  a  principle  unit, 
is  considered.  Invariants  and  covariants  of  the  algebra  are  calculated  by  the 
method  of  Section  II,  and  a  set  of  ten  of  these  functions  is  shown  to  form  a  com- 
plete system  of  invariants  and  covariants  from  the  standpoint  of  Lie. 

In  Section  IV  it  is  shown  that  for  the  example  of  Section  III  the  arithmetic 
invariant  denoting  the  rank  can  be  replaced  by  a  rational  integral  covariant. 
The  generic  case  is  defined  as  the  case  for  which  certain  three  covariants  are 


*  Presented  to  the  Society,  March  25,  1921. 
t  These  Transactions,   vol.  19  (1918),  p.  408. 

(135) 


136  C.  C.  MACDUFFEE  [March 

different  from  zero,  and  this  case  is  reduced  by  rational  transformations  in  the 
general  field  to  a  canonical  form.  The  parameters  of  the  canonical  form  are 
characterized  by  invariants. 

II.  Invariants  and  vector  covariants 

1.  Notations.     Consider  the  general  linear  algebra  in  n  units  whose  multi- 
plication table  is  given  by 

(1)  eiCj  =  YlkZ"  yuk^k  {i,  /  =  1,  .  .  .,  m), 

and  whose  general  number  is  x  =  ^'il"xiei,  where  the  yjj^  and  Xj  are  num- 
bers of  a  general  field  F.     Consider  the  transformation  of  units 

(2)  T:  e/   =  J^fJl  a^^ej    Z?  ^  |  a,;,- 1  ?^  0    .  (i  =  1,  . .  . ,  «), 

also  with  coefficients  a,;,-  in  F.  This  transformation  T  induces  upon  the  con- 
stants 7y-4  of  multiplication  and  the  coordinates  Xi  a  transformation  5  which 
carries  7,;,^  into  yijk  and  Xi  into  x,  in  such  a  way  that 

(3)  eWj  =  ^IZ"  y'ijke'k, 
and 

(4)  E:"%  =  ^  =  ^'=Z':i^'^^ 

According  to  Professor  Hazlett,  *  a  vector  covariant  is  defined  to  be  a  function 
of  the  units  and  the  constants  of  multiplication  and  the  coordinates  of  the  form 

'vilijk;  x/yBs)  (i,  j,k,r,s  =  \,  ...,  n), 

such  that,  under  a  transformation  T  of  the  units  and  its  induced  transformation 
S,  there  exists  a  relation  of  the  form 

(5)        v'  =  v{y'ijk;  xl;  e's)  =  D"  v{y,jk;  x/,  e^         {i,  j,  k,  r,  s  =  1,  . . .,  n). 

The  integer  n  is  called  the  weight  of  the  vector  covariant  v. 

As  Miss  Hazlett  noted,  f  it  follows  readily  from  (1)  and  (3)  that  every  vector 
covariant  is  expressible  linearly  in  the  units,  i.  e., 

(6)  ^'  =  T^:^ie:.     v  =  Y:rjiViei, 

where  v'i  is  the  same  function  of  the  y'ij^  and  x'^  that  f,  is  of  the  7,;,^  and  the  x,. 

*  These  Transactions,  vol.  19  (1918),  p.  408. 
t  Ibid.,  p.  €16. 


1922]  UNBAR  ALGEBRAS  137 

2.  Theorem  1.  The  coefficients  Vj  of  every  vector  covariant  of  weight  n  expressed 
in  the  form  v  =  SjC"  ^,-^,-  are  transformed  cogrediently  apart  from  the  factor  D~'' 
mith  the  coordinates  Xi  of  the  general  number  x  =  2)~"xj^,-  under  a  linear  trans- 
formation of  determinant  D. 

From  (5)  and  (6)  we  have 

Making  use  of  (2),  we  find  that 

On  account  of  the  linear  independence  of  the  units  Cj,  which  occur  only  where 
shown  explicitly, 

(7)  v;=  D-"  2];:>.'  «.>•         •         {j  =  !,:..,  ft). 

Similarly  from  (4)  and  (2)  we  have 

(8)  Xj  =  YPlxla^j  (/=!,...,«). 

Comparing  (7)  with  (8),  we  see  that  the  theorem  is  proved. 

3.  Hypercomplex  determinants.*  On  account  of  the  fact  that  multiplication 
is  usually  neither  commutative  nor  associative,  a  determinant  whose  elements 
are  hypercomplex  numbers  must  be  defined  more  precisely  than  a  determinant 
whose  elements  are  ordinary  numbers.  We  define  the  general  hypercomplex 
determinant  of  the  nth  order 


(9)  •  D  = 


an  ...  ai„ 

ani  ■  ■  ■  a„„ 
[oiia22  • . .  a„„] 


to  be  the  sum  of  n !  terms  of  the  type 

(-1)*  [ai,-,  02,,  •••  a„ij 

in  which  ii,  4 i„is  an  arrangement  of  1,  2,  . .  .,n  derived  from  the  natural 

order  by  k  interchanges.  The  first  subscripts  must  occur  in  their  natural  order 
in  every  term.  The  brackets  indicate  that  the  method  of  grouping  the  factors 
is  arbitrary,  but  the  same  method  is  to  be  used  in  each  term.     A  particular 

*  Determinants  whose  elements  are  quaternions  and  hence  associative  but  not  commuta- 
tive were  considered  by  Cayley,  On  certain  results  relating  to  quaternions,  The  Philosoph- 
icalMagazine,  vol.  26  (1845)  pp.  141-145;  and  also  by  C.  J.  Joly,  second  edition  of 
Hamilton's  Elements  of  Quaternions,  vol.  2,  Appendix  1,  p.  361. 


138 


C.  C.  MACDtnPFEE 


[March 


hypercomplex  determinant  of  the  nth  order  is  obtained  by  replacing  the  brackets 
inclosing  the  leading  term  by  a  particular  grouping  of  that  term,  in  which  case 
it  is  understood  that  every  term  of  the  expansion  is  to  be  grouped  in  that  way. 
Thus  there  are  as  many  particular  hypercomplex  determinants  of  the  nth  order 
as  there  are  ways  of  grouping  n  ordered  factors. 

4.  Lemma  1.  Every  hypercomplex  determinant  merely  changes  sign  upon  the 
interchange  of  any  two  columns. 

Consider  the  determinant  (9).  Its  terms  may  be  arranged  in  gC^!)  pairs 
of  the  type 


(10) 


(-1)*   [axi^   ...    Or, 


(-1)*+'  [au. 


.  .    Qri. a,, 


where  the  two  terms  in  each  pair  differ  only  in  having  their  i,  and  i^  subscripts 
interchanged.  Since  each  is  obtainable  from  the  other  by  one  interchange  of 
subscripts,  they  are  opposite  in  sign.  Let  us  denote  D  with  its  rth  and  5th 
columns  interchanged  by  D'.  Evidently  this  interchange  leaves  the  first  sub- 
scripts in  their  natural  order  but  interchanges  the  second  subscripts  v  and  i, 
wherever  they  occur,  and  so  interchanges  the  absolute  values  of  the  terms  in 
each  pair  (10).  Thus  every  term  in  D  with  its  sign  changed  is  equal  to  a  term 
of  D'  and  vice  versa.  Then  D'  =  —D.  This  argument  depends  in  no  way 
upon  the  manner  of  grouping,  so  the  lemma  holds  for  all  hypercomplex  deter- 
minants. 

No  analogous  theorem  exists  concerning  the  interchange  of  two  rows. 

Lemma  2.  Any  hypercomplex  determinant  two  of  whose  columns  are  identical 
is  zero. 

For  by  the  interchange  of  the  two  identical  columns  the  determinant  is  un- 
altered, and  yet  changes  in  sign. 

5.  Lemma  3.  A  hypercomplex  determinant  the  elements  of  whose  jth  column  are 
binomials  a^j  -\-  bij  is  equal  to  the  sum  of  two  determinants  identical  with  the  first 
except  that  the  jth  column  of  the  one  is  composed  of  the  a,j-  while  the  jth  column  of  the 
other  is  composed  of  the  b^j.     The  corresponding  theorem  holds  for  rows. 

Let  us  set 


D  = 


an   •  •  •    ihr  +  Clr)    ■  ■  ■   flln 


a»l   •  •  •    {bnr  +  O    •  •  •    flnx 


A   = 


ail    ■  ■  ■   i>lr  ■  ■  •   (^In 

Obi  •  •  •  b„r  . .  ■  a„„ 


A  = 


On    ...    Clr    ...    Oin 


flnl   •  •  •   C„r  .  .  .   a„ 


1922] 


LINBAR  ALGSBRAS 


139 


where  it  is  understood  that  the  manner  of  grouping  is  arbitrary,  but  the  same 
manner  must  be  used  consistently  in  every  term  of  each  determinant.  It 
follows  from  the  distributive  law  that 


(11)  a„- 


(Psi^  +  Csi)  . . .  a„i^  =  au^  . . .  bsi^ 


+  au  ...  Cs, 


for  all  values  of  the  subscripts  and  for  all  methods  of  grouping.  It  holds  in 
particular  for  i^  =  r  and  5  =  1,  . . .,  n  where  r  is  fixed.  Let  k  denote  the  num- 
ber of  interchanges  necessary  to  obtain  the  order  ij,  . . . ,  i„  from  the  natural 
order  1,  . .  .,  n.  Then,  multiplying  (11)  by  (  —  1)  and  summing  for  ii,  . . .,  i„, 
we  have 

D  =  Di  +  Di, 

which  was  to  be  proved. 

We  prove  the  corresponding  theorem  for  rows  by  noting  that  (11)  holds  when 
5  is  fixed  and  i^  ranges  over  the  values  1,  . .  .,n,  and  summing  as  before. 

6.  Theorem  2.     The  determinant 


V  = 


ei  ...  e„ 


ei 


is  a  vector  covariant  of  weight  1  of  the  linear  algebra  in  n  units  ei, 
manner  of  grouping. 
We  set 

V'  = 


Under  transformation  (2)  this  becomes 


. ,  e„  for  every 


V 


^:;:>i,.,  ...^;::>„,,.., 


Z';:r«iu^^....Z'::r««^n^.. 


From  Lemma  3  we  see  that 

V,  = 


<i »,=i 


"i,-,  e,^ 


«n.-„  ei. 


"ifi  «.i 


By  Lemma  2  such  of  these  determinants  as  have  two  identical  columns  are  zero. 
Moreover  the  as  are  numbers  of  the  field  F.  Hence  we  may  restrict  the  I'l, 
. . . ,  t„  to  sets  of  n  distinct  values  and  write 


140 


C.  C.  MACDtTFFBE 


[March 


F' =  !:«:,.. 


•  ei„ 


e„  ...  e,. 


Let  k  denote  the  number  of  interchanges  necessary  to  produce  the  order  j'l, 
. .  .,i„  from  the  natural  order  of  the  integers  1,  . .  .,n.  By  Lemma  1  the  above 
determinant  changes  sign  with  each  such  interchange,  that  is  with  each  inter- 
change of  two  columns.     Then 

€t    ...    € 

V'^J2i-'^)'au,...   a,.-.    .     ".'     ." 
'■ ■'■"  e,...e„ 

The  above  sum  is  exactly  the  determinant  D  of  the  transformation  (2).  Hence, 
as  stated, 

V  =  DV. 

7.  Rational  integral  invariants.     In  the  general  linear  algebra  in  n  units  the 
right-  and  left-hand  characteristic  determinants* 

5'('^)  =  I'^ZiyiJk^j  -  ^ik<^\        (i,  j.k  =  1,  ...,  n), 


(12) 


and  the  coefficients  of  the  powers  of  co  therein  are  absolute  covariants  under  the 
general  group  of  homogeneous  linear  transformations  on  the  units.  In  view  of 
Theorem  2,  there  are  conceivably  as  many  vector  covariants  of  weight  1  as  there 
are  ways  of  grouping  n  ordered  factors,  but  it  may  happen  in  certain  algebras 
that  a  smaller  number  results.  Thus  for  commutative  algebras  they  all  vanish 
and  for  associative  algebras  they  become  identical.  By  means  of  the  multi- 
plication table  (1)  each  such  vector  covariant  can  be  expressed  linearly  in  terms 
of  the  units,  i.  e.,  in  the  form  (6).  Suppose  that  p  of  them  are  linearly  indepen- 
dent. By  Theorem  1  the  coefficients  Vi  are  transformed  cogrediently  apart  from 
the  factor  D~^  with  the  coordinates  a;,  of  the  general  number.  Therefore  if 
these  coefficients  ^,-  be  inserted  in  place  of  the  Xj  in  a  homogeneous  absolute 
invariant  of  degree  r,  there  results  a  relative  invariant  of  the  linear  algebra  of 
weight  r.     This  method  yields  2np  relative  invariants. 


III.  A  Lie  complete  system  of  invariants  and  covariants  for 

A  TERNARY  ALGEBRA 

8.  Foreword.  It  will  be  shown  that  for  the  algebra  with  three  units,  one  of 
which  is  a  principal  unit,  the  invariants  formed  by  the  method  of  the  preceding 
section  together  with  those  obtained  from  the  characteristic  determinants  by 

*  h.  E.  Dickson,  Linear  Algebras,  Cambridge  Tract  No.  16  (Cambridge,  1914),  p.  17  and 
p.  20. 


1922]  LINEAR  ALGEBRAS  141 

the  methods  of  the  theory  of  invariants  of  forms  constitute  a  complete  system 
of  invariants  and  covariants  from  the  standpoint  of  Lie.  In  fact,  if  all  the 
invariants  obtainable  by  both  methods  are  taken,  there  is  considerable  redun- 
dance. A  set  of  the  simpler  ones  is  shown  to  be  composed  of  functionally  inde- 
pendent invariants. 

9.  The  finite  transformations.     In  the  linear  algebra  in  three  units  1,  d,  e^, 
where  1  is  a  principal  unit,  the  general  number  is  of  the  form 

(13)  X  =  Xo  -\-  Xiei  +  X2^2 

where  xo,  Xi,  xi  are  numbers  of  a  field  F.  The  multiplication  table  may  be  taken 
to  be 

<?i^  =  ao  +  ai^2  +  0262, 
Q  .X  e>^  =  bo  +  biei  +  62^2, 

^1^2    =    Co    +   Ciei    +   €202, 

eiCi  =  c?o  +  die\  -f-  d2e2, 

where  the  coefficients  are  numbers  of  F.     By  applying  the  linear  transformation 

of  units  • 

/,  c-v  ^1  =  ao  +  ociBi  -\-  a2^2, 

^     '  €2  =  Po  +  /3iei  +  ft^2, 

where  the  coefficients  are  numbers  of  F  such  that  D  ^  01182  —  «2/3i  is  different 
from  zero,  the  principal  unit  is  left  invariant  and  the  general  number  and  multi- 
plication table  become  (13)  and  (14)  respectively  with  each  letter  primed.  That 
is,  the  transformation  (15)  of  units  induces  upon  the  coefficients  the  following 
transformation : 

x'a  =  Xo  +  D-\a2&o  -  oio^2)xi  +  D-^(aopi  —  a;i/3o)^2, 
x[  =  D-i  ^a-Ti  -  £>-»  fiiX2, 
X2  =  —  D~^  UiXi  +  D~^  aiXi, 

a'o=  A+  D--^{a2po  -  "oft)^  -f-  D-^aoft  -  ai/3o)C, 
a[  =  Z?-i  P2B  -  D-^fiiC, 
a'2  =   -D-i  ajS  -f-  Z)-i  aiC, 
(16)  ^'o    =  -E  +  D-^(a20o  -  ao02)F  +  r>-i(aoft  -  a,0o)G, 

b[   =  Z?-i  ftF  -  D-i  ^iG, 
b!,   =   -  Z?-i  a2F  +  Z?-»  aiG, 

c'„    =  H  +  D-^{a20o  -  aoffi)!  +  D-'{ao0i  -  a,0o)J, 
c[    =  D-^  02l  -  -D-i  0ij, 
C2    =   -  D-^  a2l  +  D-'  aj, 

d'o  =  K  +  D-\a2^o  -  ao02)L  +  Z?-i(aoft  -  ai(3o)M, 
d[  =  D-»  ^L  -  D-i  /3iM, 
d'2  =  -  D-^a2L  +  D-'aiM, 


142  C.  C.  MACDUFFEE  [March 

where 

A  =  ao^  +  ai^'ao  +  02^60  +  cL\aiPa  +  aias^o, 

B  ^  a-^a\  +  a-^h\  +  2aoai  +  a\aif:\  +  ctiai^x, 

C  ^  01.^0.2  +  oi^hi  +  2aoa2  +  aia2C2  +  «i«2rf2> 

£  =  ^0'  +  /Si^ao  +  fii'ba  +  /Si^s^o  +  &iM<>, 

F  =  |3i=ai  +  ^2^61  +  2/3o|3i  +  /Siftci  +  /SiMi, 

6^      =    /3i%2   +    ^^h   +   2/3o;32   +    PACi   +    fiyl3^2, 

H  =  ao/3o  +  ai/SiCo  +  02/3260  +  aiftco  +  a2/3ido, 

7  =  ai/3o  +  ao/3i  +  ai/SiOi  +  02/3261  +  ai/32Ci  +  a20idi, 

J  =  a2/3o  +  ao/32  +  ai/3ia2  +  02/3262  +  ai/32C2  +  a^ffich, 

K  =  Qo/3o  +  oi/Jiao  +  02/8260  +  a2/3iCo  +  ai^^o, 

L  =  ao/3i  +  oi/3o  +  oijSiOi  +  02/3261  +  02/3iCi  +  oi/32(ii, 

M  =  00P2  +  02180  +  ai/3ia2  +  02/8262  +  02/3iC2  +  ai^zdi. 

10.  The  Lie  group.  The  generators*  of  the  infinitesimal  transformation 
corresponding  to  the  finite  transformation  (16)  are  found  to  consist  of  six  partial 
differential  equations  which  it  is  not  necessary  to  give  here  in  detail.  The 
first  term  of  each  equation  involves  only  the  Xi,  and  is  the  only  term  in  which 
the  Xi  occur.  Since  these  equations  are  generators  of  an  infinitesimal  group 
corresponding  to  the  finite  group  of  transformations  (16),  they  form  a  fortiori 
a  complete  system  of  partial  differential  equations.  The  six  equations  in  fifteen 
variables  have  nine  functionally  independent  solutions,  and  these  solutions  form 
a  complete  set  of  absolute  invariants  and  covariants  of  the  linear  algebra  (14). 

Moreover,  if  the  terms  involving  Xo,  Xi,  X2  in  these  equations  be  deleted,  there 
results  another  complete  system  of  partial  differential  equations,  the  generators 
of  the  group  corresponding  to  the  group  of  transformations  in  (16)  on  the  con- 
stants of  multiplication  only.  This  system  has  six  functionally  independent 
solutions,  the  six  absolute  invariants  of  the  linear  algebra.  They  are  six  of  the 
nine  solutions  of  the  first  complete  system.  We  see  then  that  there  are  exactly 
six  absolute  invariants  and  three  absolute  covariants  in  a  complete  system  of 
invariants  and  covariants  for  the  algebra.  There  are  then  no  more  than  seven 
functionally  independent  relative  invariants.  A  complete  system  will  be 
exhibited  of  three  absolute  covariants  and  seven  relative  invariants  whose 
jacobian  does  not  vanish  identically. 

11.  Invariants  of  the  characteristic  determinants.  The  right- and  left-hand 
characteristic    determinants    (12)    become 

5(co)  =  -  o)'  -h  Aoj^  -  Tco  -f-  A, 
5'(a))  =   -  0)5  -f-  A'co"  -  r'w  -f-  A', 


•  Lie-Scheffers,  Vorlesungen  iiber  continuierliche  Gruppen,  Leipzig,  1893,  p.  716,  et  seq. 


1922]  LINEAR  ALGEBRAS  143 

where 

A  =  Sjco  +  (ai  +  ct)xi  +  {di  +  b^X2, 

r  =  Zxi?  +  {ayCi  —  a2Ci  —  ao)xi^  +    (Wi  —  hdi  —  60)^2^  +  (2ai 
+  2c^Xiix-i.  +  (262  +  2di)xtiXi  +  (0162  —  02^1  +  c-4i  —  Cid2  —do  —  Co)xiXt, 

(17)  A   =  a;o'  +   (02^0  —  aoC2)xi'  +   (6i(io  —  Mi)«2'  +  (ciiCi  —  OjCi 

—  Co)  xoxr^  +   (ai  +  C2)xo^Xi  +  (62  +  c/i)x:o^X2  +  (Wi  —  hd^  —  60)  xoXi^ 
+  (coC?2  —  Cido  +  02^0  —   00^2  +  OoCi  —  aiCo)xi''x2  +  (60^2  —  Wo  +  Qo^i 

—  aibo    +   CiC?o    —    Codi)xix^'^    +    {c^di    —    cA    +  aib2   —    0261    —    co 

—  do)^0^1^2> 

and  A',  V,  A'  are  obtained  from  A,  T,  A  respectively  by  interchanging  c,  and 
d,-,  for  i  =  0,  1,  2.  Each  of  these  six  expressions  satisfies  the  equations  of  §10 
and  is  an  absolute  covariant  of  the  algebra. 

The  invariants  of  the  six  ternary  forms  (17)  are  invariants  of  the  algebra. 
A  and  A'  have  no  invariants.  F  and  T'  have  one  invariant  each,  their  hessians. 
The  ternary  cubic  forms  A  and  A'  each  have  two  relative  invariants,  the  5  and 
T  of  Aronhold.*  But  only  six  of  the  seven  invariants  have  been  accounted  for 
by  this  means,  even  if  all  six  should  prove  to  be  independent.  Thus  it  is  evident 
that  the  invariants  obtainable  as  the  invariants  of  the  coefficients  of  the  char- 
acteristic determinants  are  not  sufficient  to  form  a  complete  system. 

12.  Additional  invariants  by  the  method  of  Section  II.  By  Theorem  2  of  §6, 
the  hjrpercomplex  determinant 


1 

ei 

€2 

1 

e\ 

62 

1 

ei 

e2 

V  ^   1     e\    €2   =  ^1^2  ~  B2P1 
1     ei    €2 

is  a  vector  covariant  of  weight  1.  By  the  multiplication  table  (14)  this  can  be 
expressed  as 

(18)  F  =  (co  -  rfo)  +  (ci  -  d,)ei  +  (C2  -  ^2)^2- 

Now  by  Theorem  1,  §2,  the  coefficients  of  the  units  in  (18)  are  transformed  co- 
grediently  apart  from  the  factor  D~^  with  the  coefficients  Xo,  Xi,  X2  of  the  general 
number,  and  hence  when  substituted  for  these  variables  in  (17)  give  relative 
invariants  of  the  algebra. f    Thus  we  have 

Ai  =  3(co  —  do)  +  (ai  +  C2)(ci  —  di)  +  (di  +  62)  (^2  —  £^2), 

A[  =  3(co  -  do)  +  (fli  +  d2)(,ci  —  di)  +  (ci  +  62)  (C2  -  £^2), 

r2  =  3(co  —  do)^  +  (aiC2  —  a2Pi  —  ao)(ci  —  diY  +  •  ■  •  , 

Tj  =  Z{co  -^  doY  +  (aA  —  a^i  —  ao){c\  —  diY  +  ■  •  • , 

A3  =  (co  —  doY  +  (02^0  —  aoC2)(ci  —  diY  +  ■  ■  ■  , 

A3  =  (co  —  doY  +  (.Oido  —  aod2){ci  —  diY  +  •  •  •  . 


(19) 


*  Salmon,  Higher  Plane  Curves,  Dublin,  1897,  pp.  191-192. 

t  Throughout  this  article  the  subscript  on  the  symbol  for  an  invariant  indicates  its  weight. 


144  C.  C.  MACDUFFEE  [March 

Each  of  these  forms  is  transformed  into  a  function  of  itself  by  the  differential 
operators  of  §10. 

Evidently  there  are  functional  relations  between  the  twelve  relative  invariants 
which  these  methods  yield,  since  it  has  been  shown  that  there  can  be  but  seven 
functionally  independent  invariants.  This  redundance  makes  unnecessary 
the  use  of  the  complicated  5  and  T  invariants  of  A  and  A'.  The  5  invariant  of 
A  —  A'  is  quite  simple  however,  and  will  be  used. 

13.  A  complete  system  of  invariants.  If  will  be  shown  that  the  following 
three  covariants  and  seven  invariants  form  a  complete  system : 

(20)  A,  A',  e  =  r - r',  H^  =  hessian  of  V,  H^'  =  hessian  of  T', 

54  =  5  of  A  -  A',  Ai,  Tj,  r/,  A3. 

Since  we  have  only  ten  relations  in  -fifteen  variables,  it  is  sufficient  to  show 
that  they  are  independent  when  five  of  these  variables  are  put  equal  to  constants. 
It  is  found  convenient  to  set  ao  =  ai  =  C2  =  t/s  =  0  and  a2  =  I.  In  fact  this 
normalization  can  be  made  upon  T,  V,  A,  A'  before  H2,  H2',  and  Si  are  calculated. 
The  ten  invariants  (20)  then  reduce  to  the  fairly  simple  forms: 

A  =  3:1^0  +  (^2  +  di)xi, 

A'  =  Zxa  +  (62  +  Ci)x2, 

9  =  —  (ci  —  di)xi^  —  biici  —  ^1)^2'  —  2(ci  —  di)xQX2, 

H2  =  4ci{di^  -  Ml  +  62^)  +  Uboci  -  3bi^  -  66i(co  +  d,) 

-3(co  +  do)^ 

Hi'  =  4di(ci^  -  biCi  +  62')  +  12bodi  -  3bi^  -  66i(co  +  do) 

(21)  -  3  (Co  +  d^y, 

Si     =  (ci  -  diY/Sl  [3bi{co  -  doY  +  6{ca  -  d„){c,d^  -  c^dO 

+  3b,  (Co  -  do){ci  -  di)  -  {ci  -  diy  (V  +  3bo)l 
Ai      =  3(co  —  do), 
T2     =  3(co  -  doY  -  c-i(ci  -  dO\ 
T2'    =  3{co  -  do)'  -  di(ci  -  d,Y, 
A3     =  (co  —  do)^  +  co(ci  —  di)^  —  Ci(co  —  do){ci  —  di)'^, 

14.  Independence  of  the  invariants.  To  prove  the  independence  of  these 
ten  invariants  it  is  sufficient  to  prove  that  the  jacobian 

(22)  d(A,  A',  e,  H2,  Hi',  Si,  Ai,  T^,  T^',  A3) 

i>{xo,  xi,  Xi,  bo,  bi,  bi,  Co,  Ci,  do,  d\) 

does  not  vanish  identically.  Now  only  the  first  three  of  these  polynomials  (21) 
involve  xo,  Xi,  X2,  and  only  the  first  six  involve  bo,  bi,  bi.  Hence  it  follows  from 
considering  the  Laplace  development  that  the  jacobian  (22)  factors  into  three 
factors,  viz., 


1922]  LINEAR  ALGEBRAS  145 

d(A,  A^  e)  ^  bjHj,  Hi',  Si)  ^  a(Ai,  Fa,  ^2^  A,) 

b{xo,  xi,  X2)         d(6o,  61,  62)  d(co,  ci,  rfo,  ^1) 

It  is  sufficient  to  show  that  no  one  of  these  three  jacobians  vanishes  identically 
in  ci,  and  hence  it  is  sufficient  to  show  that  the  coefficient  of  the  highest  power  of 
Ci  in  each  jacobian  is  not  identically  zero.  It  is  then  permissible  to  drop  all 
terms  in  each  element  except  those  involving  Ci  to  the  highest  power  to  which 
it  occurs  in  that  element;  for  the  other  terms  evidently  cannot  enter  into  the 
term  of  highest  degree  in  Ci  in  the  expansion  of  the  determinant.  It  is  important 
to  be  sure,  however,  that  the  coefficient  of  the  highest  power  does  not  vanish, 
for  this  method  does  not  give  the  coefficients  of  lower  powers  correctly.  By 
this  method  it  is  easy  to  show  that 


d(A,  A',  e)        „      ,    ,   ,  , 
■■ '  =  oxict'  +  lower  powers  of  Ci, 

d(Xn.  X^.  X-)) 


b{Xo,  Xu  Xi) 


oiHi,  Hi  ,  Si)  16     ,  ,  I  \     s    1    1  r 

= Oi(co  —  da)  Ci^  +  lower  powers  of  Ci, 

i>(bo,b„bi)  9 

d(Ai,  r^,  Ti',  A3)  ^  _  g^^,  ^  j^^^^  p^^^^^  ^j  ^^_ 

d(co,  ci,  do,  di)  ■ 

Then  the  jacobian  (22)  becomes 

96  Xidi{co  —  do)ci'^*  +  lower  powers  of  Ci, 

and  the  ten  polynomials  (20)  are  functionally  independent  and  form  a  complete 
system  of  invariants  and  covariants  of  the  linear  algebra  (14)  from  the  stand  - 
point  of  Lie. 

IV.  Characterization  by  invariants  of  a  canonical  form 

15.  The  rank  covariant.  Let  us  consider  the  algebra  in  three  units  1,  d,  et 
whose  general  number  and  multiplication  table  are  given  by  (13)  and  (14) 
respectively.  The  rank*  of  every  such  algebra  is  three  or  two  according  as  every 
number  does  not  or  does  satisfy  a  quadratic  equation.  This  rank  is  an  arith- 
metic invariant  under  every  linear  transformation  of  units  (15).  It  will  be  shown 
that  for  this  example  the  arithmetic  rank  invariant  can  be  replaced  by  a  rational 
covariant. 


*  There  is  an  equation  p(oi)  =  0  of  lowest  degree  having  X  as  a  right-hand  (left-hand) 
root.  The  degree  of  this  equation  is  called  the  right-hand  (left-hand)  rank  of  the  algebra. 
(Cf.  Dickson,  Linear  Algebras,  p.  23.)  It  can  be  shown  that  there  is  at  least  one  number 
satisfying  no  equation  of  lower  degree. 


146  C.  C.  MACDUFFEE  [March 

In  every  algebra  of  rank  3  with  a  principal  unit,  there  is  some  number  x 
which  does  not  satisfy  a  quadratic  equation.  Then  the  powers  1,  x,  x"^  are 
linearly  independent  and  may  be  taken  as  the  units  1,  ei,  e^.  That  is,  the  multi- 
plication table  of  every  rank  3  algebra  in  these  units  has  the  form  (14)  with 
Oo  =  ai  =  0,  02  =  1.  Conversely,  every  algebra  of  this  form  is  of  rank  3. 
Evidently,  then,  a  necessary  and  sufficient  condition  that  an  algebra  (14)  be  of 
rank  3  is  that  it  be  possible  to  make  the  above  normalization. 

From  equations  (16)  it  is  seen  that  the  conditions  on  a  transformation  (15) 
which  shall  make  ao'  =  a/  =  0  and  02'  =  1  are 

A  +  D-\a20o  -  ao02)B  +  D-'iao0,  -  a^C  =  0, 
D-'ftS  -  D-ij3iC  =  0, 
.-D-^UiB  +  D-^aiC  =  1. 

These  conditions  are  readily  found  to  be  equivalent  to 

(23)  /3o  =  A,     ■    ^1  =  B,         /32  =  C, 

where  A,  B,  C,  given  in  (16),  are  polynomials  in  the  as  and  the  constants  of 
multiplication.  Thus  the  as  can  be  chosen  arbitrarily  and  the  P's  calculated 
from  the  above  relations,  provided  only  that  the  determinant  of  the  transforma- 
tion 

(24)  D  =  Oztti'  +  (—  fli  +  C2  -f  62)01^02  +  (62  —  Ci  —  di)aiai^  —  bia^^ 

be  different  from  zero.  Now  a\  and  a^  can  evidently  be  chosen  so  that  D  is 
not  zero  unless  every  coefficient  of  D  vanishes.  Then  a  necessary  and  sufficient 
condition  that  the  algebra  be  of  rank  3  is  that  not  every  coefficient  02  etc.  of 
(24)  vanish. 

It  will  now  be  shown  that  there  exists  a  covariant  $  whose  coefficients  are  the 
coefficients  of  (24).     From  (17)  we  form  the  absolute  covariant 

r'  -  r  +  I  (A2  -  A'2) 

every  term  of  which  involves  (ci  —  <fi)  or  {d  —  d^).  It  was  shown  in  (18)  that 
(co  —  da),  (ci  —  dC},  (c2  —  d'i)  are  transformed  cogrediently  apart  from  the 
factor  D~^  with  xq,  xi,  Xi,  so  we  may  substitute  X\  and  xi  respectively  for  these 
expressions  and  obtain  a  relative  invariant  of  weight  —  1, 

$  =  a^-^  +  (—  fli  +  C2  +  d2)x^X2  -\-  (62  —  Ci  —  di)xxx^  —  h\X^, 

whose  coefficients  are  the  coefficients  of  (24).  Hence  a  necessary  and  sufficient 
condition  that  an  algebra  (14)  be  of  rank  3  is  that  $  does  not  vanish  identically. 


1922]  LINEAR  ALGEBRAS  147 

16.  Determination  of  a  canonical  form.  We  shall  designate  as  the  generic 
case  that  case  for  which  neither  $,  r2  —  Tj',  nor  A  —  A'  vanishes  identically, 
and  reduce  this  generic  case  by  rational  transformations  in  the  general  field  F 
to  a  canonical  form — i.  e.,  to  a  form  in  which  each  remaining  coefficient  is  a 
relative  invariant  under  the  most  general  transformation  which  does  not  destroy 
the  normalization.  Since  $  is  not  identically  zero,  the  multiplication  table  can 
at  once  be  reduced  as  in  §15.  Then  from  (23)  we  see  that  in  order  to  preserve 
this  normalization  the  ^'s  must  obey  the  conditions: 

^0  =  "0^  +  ai^ba  +  aia^ica  +  d^), 

(25)  /3i  =  ai'bi  +  2aoai  +  aia^{ci  +  d,), 

/32  =  ai^  +  0:2^62  +  2aoa2  +  a\ai{ci  +  ^2). 

If  C2  —  di  is  not  already  zero,  it  is  possible  to  effect  a  transformation  making 
Ci    —  di   equal  to  zero.     From  (16)  it  is  seen  that  under  every  transformation, 

(26)  C2'  -  J2'  =   -  aiid  -  di)  +  ai(c2  -  di). 

Therefore  if  we  choose  ai  and  0:2  satisfying  the  conditions 

/g^N  «i('^2  ~  ^2)  =  a^ici  —  di), 

D  =  ai'  +  (c2  +  di)ai^a2  +  (62  —  Ci  —  di)aia2^  —  bia2^  7^  0, 

and  the  /3's  according  to  (25),  we  have  the  required  transformation.  Multiply- 
ing the  second  equation  of  (27)  by  (c2  —  ^2)'  and  eliminating  ax  by  means  of  the 
first  equation,  we  have 

D  =  ai'[(Cl   -  d,y  +  (C2  +  d2){C2  -  d^)(cx  -  d,Y 

+  (62  -  61  -  Ci  -  d,){c2  -  d,Y]. 

Hence  this  normalization  is  possible  if  and  only  if  the  expression  in  brackets  is 
different  from  zero.     But  this  expression  is  precisely  the  reduced  form  of  the 
invariant  r2'  —  r2  which  we  have  assumed  not  zero. 
Since  the  covariant 

A  —  A'  =  (c2  —  dijxi  —  (ci  —  dijXi 

does  not  vanish  identically  for  this  case,  we  know  that  Ci  —  di  is  not  zero.  From 
(26)  it  is  then  seen  that  a  necessary  and  sufficient  condition  on  a  transformation 
that  shall  leave  this  normalization  undisturbed  is  that  a2  =  0.  Then  condi- 
tions (25)  require  that  ai  7^  0,_  0:2  =  0,  j3o  =  ao^  .81  =  2aoai,  ft  =  ai^,  and 
hence  D  =  ai'.     We  see  from  (16)  that  by  (15) 

(28)  C2'  =  3aoai  -f-  ai^i. 


148  C.  C.  MACDUFFEE  [March 

SO  let  US  now  apply  the  transformation 


/ao     ai     ajN         / —Ci/3         1  0\ 

Uo     iSi     ft/  "  V  C2V9    -2c2/3     1/ 


which  reduces  C2'  to  zero.     Dropping  accents,  we  obtain  the  canonical  form  of 
the  generic  case  for  rational  transformations,  viz., 


ei^ 

= 

e2, 

e-? 

= 

bo 

+  biei  +  62^2, 

exBi 

= 

Co 

+  CiCi, 

e^ei 

= 

do 

+  diei, 

(29) 

ci  -  di  ^  0. 

It  is  evident  from  (28)  that  for  C2  =  0  and  ai  9^  0,  a  necessary  and  sufficient 
condition  that  C2'  =  0  is  that  ao  =  0.  Then  the  most  general  transformation 
which  will  leave  (29)  unaltered  in  form  is  of  the  type 


/O     ai      0  \ 

(0     0     J'^  = 


The  effect  of  this  transformation  on  the  coefficients  of  (29)  is  to  make 

(30)  bo'  =  ai%,     bi'  =  ai^bu       b^     =    ai^bi,     Co     =   ai'co,    c/    =    aiVi, 

do'  =  aiHa,    d\    =  ai^d\. 

Hence  no  further  reduction  by  rational  transformations  is  possible,  although 
by  a  transformation  in  general  irrational  any  one  of  these  coefficients  which  is 
not  zero  can  be  reduced  to  unity.  Then  the  other  six  are  parameters  which 
cannot  be  altered  except  by  a  factor  which  is  a  root  pf  unity. 

17.  Characterization  of  parameters.  It  is  seen  from  (30)  that  the  most  that 
any  transformation  can  do  to  (29),  preserving  its  form,  is  to  multiply  each  co- 
efficient by  a  power  of  the  determinant.  It  is  natural  to  expect  then  that  each 
coefficient  in  (29)  is  the  reduced  form  of  a  relative  invariant  of  the  original  form 
(14).  We  shall  expect  co,  do  and  bi  to  be  relative  invariants  of  weight  1 ;  Ci,  di 
and  62  of  weight  | ;  and  60  of  weight  |.  We  prove  that  this  is  so  by  actually 
finding  these  invariants  in  terms  of  known  invariants.  The  invariants  which 
characterize  co,  ci,  do,  di  will  be  denoted  by,  respectively,  C,  C,  D",  D'. 

Making  the  normalization  indicated  in  (29),  we  find  that  the  invariants  (19) 
reduce  to  the  forms 

Ai  =  Ai'  =  3(co  —  do), 

,„..  T2  =  3(co  -  ^0)==  -  ci(c,  -  d,y, 

^^^^         .    r^'  =  3(<:„  -  do)'  -  d,{c,  -  di)\ 

A3  =  A3'  =  (co  —  do)^  +  ca{ci  —  di)'  -  Ci{cq  -  do){ci  -  di)^. 


As  - 

-^A,3  +  |A:(|A,= 

-  r^) 

A3  - 

-r,') 

1922]  LINEAR  ALGEBRAS  149 

We  have  exactly  four  relations  to  determine  four  unknowns.  We  find  Co  in 
terms  of  the  reduced  invariants  (31)  and  define  C  to  be  the  corresponding  func- 
tion of  the  complete  invariants,  so  that  C  is  an  invariant  which  reduces  to 
Co  under  the  normalization  (29) .  Similarly  we  find  invariants  which  reduce  to 
the  other  parameters.    Thus,  since  Ti'  —  r2  5^  0, 


.C 


(r^'-r,)"  (r^^  -  r,)^ 

Evidently  the  weights  are  as  predicted. 

18.  Equations  defining  the  remaining  parameters.  Let  us  now  consider  the 
problem  of  finding  invariants  which  characterize  ^o,  bi,  b^.  In  addition  to  the 
invariants  (31)  we  use  the  hessians  H2  and  H^'  of  T  and  r',  and  the  invariants 
54  and  Te  of  A  —  A'.     Under  the  normalization  (29)  we  find 

ia)H,,    {b)H,',    (c)  54  as  given  by  (21), 
^  ^        (d)  Ti  =  4l^mp^  -  ZlVp^  +  QpHqrs  -  12 mrpq^  +  8gV'  -  27 p\h^, 

where 

I  =  Ca  —  do,  m  =  bo(ci  —  di)  —  6i(co  —  do),  P  =  ^{ci  —  di), 
q  =  ^(ci  -  di),  s  =  J  [62(^0  -  ^0)  +  2cido  —  2codi], 
r  =  —^bi(ci  —  di). 

Making  use  of  the  relation  (33)  in  (32c),  we  have  (34) : 

(33)  F2  -H2'  =  4(ci  -Ji)  (622  +  3^0  -ccfi); 

(34)  61  (co  -  doKci   -  d,)   +  b,{co  -  d,Y   =   i   [81  54/(ci  -  d,) 

-  6(co  -  do)  (ciJo  -  c^i)  +\{ci-  d,){Hi  -  H^')  +  Cidi{ci  -  d,)^. 

Multiplying  (326)  by  Ci  and  subtracting  this  product  from  the  product  of  (326) 
by  di,  we  obtain 

(35)  36i2  +  66i(co  +  do)  +  ^b^idx  =   {d^H^  -  CiHi')/{ci  -  d,) 

+  Ac,d,  (ci  +  dx)  -  3(co  +  do)\ 


130  C.  C.  MACDUFFEE 

Directly  from  (33)  comes 

(36)  h'  +  3bo  =    ^^^^  _  ^^^^   IH2  -  H2'  +  4cMci  -  di)]. 

Now  Co,  do,  Ci,  di  are  the  values  which  the  known  invariants  C,  D",  C,  D' 
take  when  the  general  algebra  becomes  (29).  Consider  functions  B",  B',  B' 
of  the  coefficients  of  the  general  algebra  such  that  B"  has  the  value  bo,  B'  the 
value  bi  and  B"  the  value  62  when  the  general  algebra  becomes  (29).  Then 
corresponding  to  (34),  (35),  (36),  and  (32d),  we  have 


(37) 


where 


5/(^0  _  J50)((;7'  _  D')  +  5"(C''  -  D")  =  E, 

35'2  +  65'(C«  +  £>")  +  4B"C'D'  =  F, 

B"2  +  SB"  =  G, 

76  =  4  Z^wps  -  3  Pr^pi  +  6  ^2;^^^  _  12  nirpq^  +  8  gV'  -  27  ^2g2^^ 


£  =  i  [81  S4/(C'  -  P')'  -  6(C''  -  DO)iC'Do  -  CD') 

+  \  iC'  -  b'){H^  -  H,')  +  C'D'iC  -  D')^], 
F  =  (ZJ'Fa  -  C'H2')/{C'  -  D')  +  AC'D'{C'  +  D')  -  3(C''  +  £"')^ 

^  =   (4C^  -  4P0   ^-^^  ~  ^'   +  4^'-^'(^'  -  ^')1' 
Z    =  C  -  Z?»,  w  =  Bo{C'  -  D')  -  B'iC  -  Z?"), 

•     f  =  -  i  (C  -  D'),         q   =  -  I  (C  -  D'), 

;•    =  _  I  B"(,C'  -  D'),     s   =  I  [S'XC  -  £">)  +  2C'D'>  -  2C»£)']. 

Equations  (37)  determine  the  invariants  5",  J5',  B"  uniquely.  A  practical 
method  for  determining  them  is  to  calculate  the  two  solutions  of  the  first  three 
equations  of  (37)  and  retain  only  the  one  which  satisfies  the  fourth  relation. 

Thus  the  invariants  of  the  linear  algebra  (14)  which  we  have  found  are  suffi- 
cient to  isolate  and  characterize  the  generic  case. 

The  University  of  Chicago, 
Chicago,  III. 


VITA 

Cyrus  Colton  MacDuffee  was  born  June  29,  1895,  at  Oneida,  New  York. 
He  attended  the  public  grammar  school  and  high  school  in  that  city.  In  1917 
he  was  graduated  from  Colgate  University  with  the  degree  of  B.S.  The  two 
following  years  he  spent  as  instructor  in  mathematics  at  Colgate.  He  attended 
the  summer  school  of  Cornell  University  during  the  summer  of  1917.  He  en- 
tered the  University  of  Chicago  in  June,  1919,  and  was  in  residence  for  nine 
successive  quarters.  He  attended  courses  under  Professors  E.  H.  Moore, 
L.  E.  Dickson,  G.  A.  Bliss,  E.  J.  Wilczynski,  H.  E.  Slaught,  J.  W.  A.  Young 
and  K.  Laves  of  the  University  of  Chicago  faculty,  and  during  the  summer  quar- 
ters he  attended  courses  under  Professor  M.  W.  Haskell  of  the  University  of 
California,  Professor  S.  Lefschetz  of  the  University  of  Kansas,  Professor  T. 
Hildebrandt  of  the  University  of  Michigan,  and  Professor  H.  Blumberg  of  the 
University  of  Illinois.  In  June  1920  he  received  the  S.M.  degree,  and  in  Sep- 
tember 1921  the  degree  of  Ph.D.  Since  then  he  has  been  instructor  at  Prince- 
ton University. 

The  writer  wishes  to  express  his  gratitude  to  all  the  members  of  the  mathe- 
matics department,  and  especially  to  Professor  L.  E.  Dickson  under  whose 
direction  this  thesis  was  written. 


__J^-  the  lase  date  stamped  beW. 

JUN    2   1< 

SOFeb'SOCD 


LD  21-i00m-9,>, 


47(A5702sl6)47a 


